Page 101 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

84 SLENDER STRUCTURES AND AXIAL FLOW

where ( ‘ ) = a( )/at and ( )’ = a( )/at, in which the following dimensionless system
parameters have arisen:
-
TL~
M (M + m ) ~ ~ r=-,
’= El g, El
~
n=- A L ~ KL4 I E* c= cL2
k=-
El ’ EI’ .=[E(M+m)] s’ [EI(M +rn)]1/2’
(3.71)

In general, the system dynamics will depend on all of these parameters.
If the hysteretic damping model is used, it is clear from expression (3.39) that the first
two terms of (3.70) should be replaced by

(1 + pi)Q””. (3.72)

This corresponds to solutions of (3.70) of the type ~(4, t) = Y(t)exp(iws), in which w
is either wholly real or, if complex, such that %e(w) >> 9,m(w); the hysteretic model
may thus be considered as a particular case of the viscoelastic one for which (YO = p or
a%e(W) = p, respectively. The dimensionless frequency w is related to the dimensional
circular (radian) one, f2, by

(3.73)

In the case of an end-nozzle, as discussed at the end of Section 3.3.2, the definitions of
u and /3 in (3.71) need to be modified to

M U
(3.74)

With these, the dimensionless form of equation (3.42) is identical to the appropriately
simplified equation (3.70), namely

q”” + U*Q” + 2/3’/2ulj’ + = 0. (3.75)

The usefulness of the end-nozzle emerges from the second of equations (3.74): instead of
changing pipes, one may change nozzles to alter j3, at least over a range relatively close
to the initial B for the pipe without a nozzle.

3.3.6 Methods of solution

Two methods of solution will be given: the first, due to Gregory & Paidoussis (1966a),
for the simpler, homogeneous equation of motion; the second, used by Pafdoussis (1966)
and Pafdoussis & Issid (1974), applies to the fuller, nonhomogeneous equation of motion.